On the fixed points of affine nonexpansive mappings

Abstract

Let K be a closed convex bounded subset of a Banach space X and let T:K→K be a continuous affine mapping. In this note, we show that (a) if T is nonexpansive then it has a fixed point, (b) if T has only one fixed point then the mapping A=(I+T)/2 is a focusing mapping; and (c) a continuous mapping S:K→K has a fixed point if and only if, for each x∈k, ‖(An∘S)(x)−(S∘An)(x)‖→0for some strictly nonexpansive affine mapping T.